Structures and Supports

Structures is the applied end of the subtest. Two ideas cover it: how a load shares between supports, and why triangles are everywhere.

Sharing a load between supports

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A load between two supports divides between them, and the nearer support carries more.

The split is in inverse proportion to the distances. A load twice as far from support A as from support B puts twice as much weight on B.

Reading the figure:

• Centered: the 600 pounds splits evenly, 300 each.
• Nearer the left: the left takes 450 and the right 150, a 3 to 1 split.
• Directly over the left post: the left takes all 600 and the right takes nothing.

The two reactions always add to the total load, which is the check. 450 plus 150 is 600.

Calculating the split

Use the torque balance from the levers lesson. Take moments about one support and the other support's share falls out.

A 900-pound load sits on a 12-foot beam, 4 feet from the left support.

It is 4 feet from the left and 8 feet from the right, so it is twice as far from the right. The left carries twice as much.

Split 900 in the ratio 2 to 1: 600 pounds on the left, 300 on the right.

Check: 600 plus 300 is 900. Correct, and the nearer support has the larger share as it must.

The general rule: each support's share is the load times the distance to the OTHER support, divided by the total span. For the left: 900 times 8 over 12, which is 600.

Note the crossover - the left support's share uses the distance to the right support. That is where the errors are, and the sanity check that the nearer one carries more catches it every time.

Triangles and bracing

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A four-sided frame with pinned joints can change shape without any member changing length. It folds over sideways, which is what the first panel shows.

A triangle cannot. Changing its shape requires changing the length of at least one side, and the members resist that. That is why triangles are everywhere in structures.

A diagonal brace turns a rectangle into two triangles, which is why bracing works and why you see diagonals in bridges, roof trusses, scaffolding, gates, transmission towers and the back of a bookshelf.

A question asking how to stop a frame racking wants a diagonal, and the reason is the triangle rather than the extra material.

Tension and compression

Tension pulls things apart. Compression squeezes them together.

A cable can only pull. That is its defining limitation and it is asked: a question offering a cable in compression is offering something impossible. A cable pushed simply goes slack.

A beam under load sags, and that shape tells you where the forces are: the top is squeezed into compression and the bottom is stretched into tension.

That is why an I-beam has most of its material at the top and bottom flanges and little in the middle. The flanges carry the compression and the tension; the web between them is doing far less, so removing material there saves weight without costing much strength.

The five kinds of load

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A shaft under torque carries torsional shear, and most of that stress sits near the outer surface - which is why a hollow shaft carries nearly as much torque as a solid one of the same outside diameter, for much less weight.

A shear pin is designed to break: under overload it shears cheaply and protects a costlier part, like an outboard motor's gearcase.

Stress, strain and failure

Stress is force per unit area, usually in pounds per square inch.

A steel rod of 2 square inches carries 8,000 pounds. The stress is 8,000 / 2 = 4,000 psi.

Strain is deformation per unit of original length - how much it stretched compared with how long it was.

A safety factor divides the ultimate strength to get the working load. A column that can carry 40,000 pounds with a safety factor of 4 has an ultimate strength of 40,000 x 4 = 160,000 pounds.

Fatigue is failure from repeated stress cycles, well below the load that would break a part once. Fatigue cracks start at stress concentrations - a hole, a sharp corner, a knot in a rope - which is why aircraft parts have service lives even when they look perfect.

Beams and columns

A simply supported beam (supported at both ends) with a center load bends downward in the middle, and the bending is greatest there. A cantilever is supported at one end only, and its greatest stress is at the support. The bending moment at a cantilever's support is the load times its distance: 200 pounds at the end of a 10-foot cantilever gives 2,000 foot-pounds. Double the span with the same load and the bending moment doubles.

Depth matters far more than width. A beam's bending strength grows with the square of its depth, so a joist is set on edge, not flat.

A long, slender column under heavy compression may buckle sideways before the material itself fails. A short, thick column carries more.

An arch turns its load mostly into compression, which is why masonry - strong in compression, weak in tension - builds arches.

Two sling legs share a load: lifting straight up, each leg of a two-leg sling carries half. A wider sling angle increases the tension in each leg, because more of the pull goes sideways.

Heat, rust and joints

Structures must be allowed to expand. Bridges have expansion joints, rails and sidewalk slabs have gaps, and a steel member held rigidly while it heats builds large compressive stress and can buckle. Two different metals bolted together expand at different rates, which loosens the joint over repeated temperature swings. Metal loses strength as it gets very hot.

Rust is prevented by keeping oxygen and water off the steel: paint, plating, or galvanizing with zinc, which corrodes first and protects the steel beneath. A sacrificial anode works the same way: a more reactive metal corrodes instead of the part it protects. Rusting rebar expands and cracks the concrete around it.

Load paths

Every load has to reach the ground.

A load on a floor goes into the joists, into the beams, into the columns, into the foundation, into the ground. A question about what carries a load wants the next member down that path.

Removing any link in the path is what causes a collapse, which is why a supporting wall cannot be removed without replacing its function with a beam and posts.

What you can skip

Across the 101 questions on this topic in our bank:

• Material constants. The modulus of elasticity, yield points and deflection formulas never appear.
• Section properties. The moment of inertia of a beam never appears. Know that depth counts more than width, and that the I-beam puts material where the stress is.
• Truss analysis. Trusses come up twice, both asking why a triangle is rigid or what its members carry; no question solves for a member's force.

Where people lose points

Splitting a load evenly when it is not centered.

Using the near distance rather than the far one in the share calculation.

Putting a cable in compression.

Thinking a rectangular frame is rigid without a brace.

Getting a beam's tension and compression the wrong way round. Sagging means top compressed, bottom stretched.

Forgetting the reactions must add to the load.

Work one in under a minute

A 1,200-pound load sits on a 10-foot beam, 2 feet from the right support. How much does each support carry?

It is 2 feet from the right and 8 feet from the left, so it is four times as far from the left. The right carries four times as much.

Split 1,200 in the ratio 4 to 1: 960 pounds on the right, 240 on the left.

Check: 960 plus 240 is 1,200, and the nearer support has the larger share. Both confirm the answer.

Where this leads

The load-sharing calculation is the torque balance again, and the material choices here are the subject of the last lesson in the subtest.